Solved Exercises on Percent Word Problems in Grade 7

Master percent word problems: calculating percentages, real-world applications, and problem-solving strategies through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Percent Word Problem
Exercise 1
In a class of 40 students, 60% are girls. How many girls are in the class? How many boys are there?
Definition:

Percent Word Problem: A mathematical problem presented in a real-world context that requires calculating percentages of quantities.

Method for solving percent word problems:
  1. Identify the total quantity (whole)
  2. Identify the percentage
  3. Convert percentage to decimal (divide by 100)
  4. Multiply total by decimal to find the part
  5. Verify the answer makes sense in context
Girls
40 × 0.60 = 24
Boys
40 - 24 = 16
Step 1: Identify the total quantity

Total students = 40

Step 2: Identify the percentage

Percentage of girls = 60%

Step 3: Convert percentage to decimal

60% = 60 ÷ 100 = 0.60

Step 4: Calculate the number of girls

Number of girls = Total × Decimal

Number of girls = 40 × 0.60 = 24

Step 5: Calculate the number of boys

Number of boys = Total - Girls

Number of boys = 40 - 24 = 16

Girls: 24 | Boys: 16
Final answer:

There are 24 girls and 16 boys in the class.

Applied rules:

• Decimal Conversion: Percent ÷ 100 = Decimal

• Multiplication Rule: Part = Whole × Decimal

• Complementary Rule: Part 2 = Whole - Part 1

Key Concept:

When solving percent word problems, always identify the whole (total) and the percentage. The part is found by multiplying the whole by the decimal equivalent of the percentage.

2 Discount Word Problem
Exercise 2
A store offers a 25% discount on a $120 jacket. What is the sale price? If a customer has a coupon for an additional 10% off the sale price, what is the final price?
Definition:

Sequential Percent Changes: When multiple percentage changes are applied one after another, each change is applied to the current value, not the original value.

First Discount
$120 × 0.75 = $90
Second Discount
$90 × 0.90 = $81
Step 1: Calculate the first discount

After 25% discount, customer pays 75% of original price

First sale price = $120 × (1 - 0.25) = $120 × 0.75 = $90

Step 2: Calculate the second discount

Additional 10% off the current sale price

Final price = $90 × (1 - 0.10) = $90 × 0.90 = $81

Step 3: Verify the total discount

Total discount = $120 - $81 = $39

Total discount percentage = ($39 ÷ $120) × 100% = 32.5%

First sale price: $90 | Final price: $81 | Total discount: 32.5%
Final answer:

The sale price after the first discount is $90. The final price after both discounts is $81.

Applied rules:

• Sequential Application: Apply each percentage change to the current value

• Decimal Conversion: For discounts, multiply by (1 - decimal)

• Compound Effect: Multiple discounts don't add up linearly

$120
Original
→
-25%
Discount
→
$90
After 1st
→
-10%
Coupon
→
$81
Final
3 Population Growth Problem
Exercise 3
A town had a population of 8,000 people last year. This year, the population increased by 12%. Next year, it's projected to increase by another 8%. What will the population be next year? What is the overall percentage increase from last year?
Definition:

Compound Growth: When a quantity increases by a percentage each period, and subsequent increases are calculated on the new amount rather than the original amount.

After 12% increase
8,000 × 1.12 = 8,960
After 8% increase
8,960 × 1.08 = 9,676.8
Overall increase
(9,676.8 - 8,000)/8,000 × 100% = 20.96%
Step 1: Calculate population after first increase

Population after 12% increase = 8,000 × (1 + 0.12)

Population after 12% increase = 8,000 × 1.12 = 8,960

Step 2: Calculate population after second increase

Population after 8% increase = 8,960 × (1 + 0.08)

Population after 8% increase = 8,960 × 1.08 = 9,676.8

Step 3: Calculate overall percentage increase

Overall increase = (Final - Original) ÷ Original × 100%

Overall increase = (9,676.8 - 8,000) ÷ 8,000 × 100% = 20.96%

Next year's population: 9,677 | Overall increase: 20.96%
Final answer:

The population will be approximately 9,677 next year. The overall percentage increase from last year is 20.96%.

Applied rules:

• Sequential Increases: Apply each percentage increase to the current value

• Compound Effect: Multiple increases compound rather than add

• Overall Calculation: Use original and final values for total percentage

Key Concept:

When dealing with multiple percentage changes, each change is applied to the result of the previous change, not the original amount. This creates a compounding effect.

Percent Word Problems: Rules and Methods
\(\text{Part} = \text{Whole} \times \frac{\text{Percent}}{100}\)
Basic Percent Formula
Finding Part
\(P = W \times \frac{p}{100}\)
Find part given whole and percent
Finding Percent
\(p = \frac{P}{W} \times 100\)
Find percent given part and whole
Finding Whole
\(W = \frac{P}{p} \times 100\)
Find whole given part and percent
Key definitions:

Percent: A ratio or fraction expressed as a part of 100. The symbol % means "per hundred."

Word Problem: A mathematical problem presented in a real-world context using natural language.

Part: The portion of the whole that corresponds to the given percentage.

Whole: The total amount or 100% of the quantity.

Percent word problem-solving methodology:
  1. Read Carefully: Understand what the problem is asking
  2. Identify Components: Determine what is the part, whole, and percent
  3. Convert Percent: Change percentage to decimal by dividing by 100
  4. Select Formula: Choose appropriate formula based on what's unknown
  5. Calculate: Perform the arithmetic operations
  6. Verify: Check that the answer makes sense in context
Tip 1: Look for keywords like "of", "is", "are", "what percent", "what number".
Tip 2: "Of" usually means multiply in percentage problems.
Tip 3: Draw diagrams or tables to visualize the problem.
Tip 4: Always check if your answer is reasonable in the real-world context.
Real-Life Applications: Shopping (discounts, taxes), finance (interest, loans), statistics (surveys, polls), science (concentrations, compositions).
Common Pitfalls: Confusing part with whole, forgetting to convert percent to decimal, misinterpreting the question.
Percent Word Problem Rules:

• Decimal Conversion: Divide percent by 100 to get decimal

• Multiplication Order: Decimal × Whole = Part

• Sequential Changes: Apply each percentage change to the current value

• Verification: Part ÷ Whole should equal the decimal form of the percent

Solution: Exercises 4 to 5
4 Survey Data Problem
Exercise 4
In a survey of 500 people, 45% said they prefer coffee over tea. Of those who prefer coffee, 60% drink it daily. How many people drink coffee daily? What percentage of the total surveyed drink coffee daily?
Definition:

Sequential Percentages: When one percentage is applied to the result of another percentage, creating a nested calculation.

Coffee Preferrers
500 × 0.45 = 225
Daily Coffee Drinkers
225 × 0.60 = 135
Percentage of Total
135 ÷ 500 × 100% = 27%
Step 1: Find the number of coffee preferrers

Coffee preferrers = Total surveyed × Coffee preference percentage

Coffee preferrers = 500 × 0.45 = 225 people

Step 2: Find the number of daily coffee drinkers

Daily coffee drinkers = Coffee preferrers × Daily drinking percentage

Daily coffee drinkers = 225 × 0.60 = 135 people

Step 3: Find the percentage of total surveyed

Percentage of total = (Daily coffee drinkers ÷ Total surveyed) × 100%

Percentage of total = (135 ÷ 500) × 100% = 27%

Daily coffee drinkers: 135 | Percentage of total: 27%
Final answer:

135 people drink coffee daily. This represents 27% of the total surveyed.

Applied rules:

• Nested Calculation: Apply percentages sequentially to subsets

• Proportionality: Maintain proportional relationships

• Verification: Check that nested percentages make logical sense

📊
Survey
500 people
☕
Coffee
45% of 500
⏰
Daily
60% of coffee
🎯
Result
135 people
5 Mixture Problem
Exercise 5
A chemist needs to prepare 200 grams of a solution that is 15% salt. She has a 20% salt solution and pure water. How many grams of each should she use?
Definition:

Mixture Problem: A problem involving combining substances of different concentrations to achieve a desired concentration.

Salt needed
200 × 0.15 = 30g
20% solution needed
30 ÷ 0.20 = 150g
Water needed
200 - 150 = 50g
Step 1: Calculate total salt needed

Salt needed = Total solution × Desired concentration

Salt needed = 200g × 0.15 = 30g of salt

Step 2: Calculate amount of 20% solution needed

Amount of 20% solution = Salt needed ÷ Concentration of solution

Amount of 20% solution = 30g ÷ 0.20 = 150g of 20% solution

Step 3: Calculate amount of water needed

Water needed = Total solution - Solution with salt

Water needed = 200g - 150g = 50g of water

Step 4: Verify the solution

Salt in mixture = 150g × 0.20 = 30g

Concentration = 30g ÷ 200g = 0.15 = 15% ✓

20% solution: 150g | Water: 50g | Final concentration: 15%
Final answer:

The chemist should use 150 grams of the 20% salt solution and 50 grams of pure water.

Applied rules:

• Mixture Formula: Amount of substance = Total × Concentration

• Mass Conservation: Total mass of components equals total mass of mixture

• Concentration Calculation: Final concentration = Total substance ÷ Total mass

Comprehensive Summary: Percent Word Problems
\(\text{Part} = \text{Whole} \times \frac{\text{Percent}}{100}, \quad \text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100\)
Core Percent Word Problem Formulas
Core Definitions:

Percent Word Problem: A mathematical problem presented in a real-world context that requires calculating percentages of quantities.

Sequential Percent Changes: When multiple percentage changes are applied one after another, each change is applied to the current value.

Mixture Problems: Problems involving combining substances of different concentrations to achieve a desired concentration.

Percent Word Problem-Solving Steps:
  1. Read Carefully: Understand the problem context and what's being asked
  2. Identify Components: Determine the whole, part, and percent in the problem
  3. Convert Percent: Change percentage to decimal by dividing by 100
  4. Select Formula: Choose appropriate formula based on what's unknown
  5. Set Up Equation: Write the mathematical relationship
  6. Solve: Perform calculations to find the unknown
  7. Verify: Check that the answer makes sense in context
Quick Tip: "Of" means multiply, "is" means equals in percentage problems.
Memory Aid: The whole is the base amount that represents 100%.
Strategy: Draw diagrams or tables to visualize complex problems.
Verification: Always check that your answer is reasonable in the real-world context.
Real-Life Applications: Shopping (discounts, taxes), finance (interest, loans), science (concentrations), surveys (polls), demographics (population changes).
Common Scenarios: Sales, investments, grades, mixtures, commissions, tips, taxes, statistics.
Key Rules and Properties:

• Decimal Conversion: Percent ÷ 100 = Decimal equivalent

• Multiplication Rule: Decimal × Whole = Part

• Sequential Changes: Apply each percentage change to the current value

• Verification: Part ÷ Whole should equal the decimal form of the percent

• Mass Conservation: In mixture problems, total mass is conserved

🔍
Read
Understand the problem
📋
Identify
Find key information
🧮
Calculate
Apply formulas
✅
Verify
Check the answer

Questions & Answers

Question: I get confused about which number to use as the whole and which as the part. How do I figure this out?

Answer: Great question! The key is to identify what represents 100% in the problem:

Whole: The total amount that represents 100%. It's usually the larger number or the total quantity mentioned.

Part: The portion of the whole. It's usually smaller than the whole.

Keywords to look for:

  • "60% of 40 students" → 40 is the whole, ? is the part
  • "What percent of 50 is 15?" → 50 is the whole, 15 is the part
  • "15 is 25% of what number?" → 15 is the part, ? is the whole

The whole is usually what comes after "of" in the sentence. The part is what the percentage represents.

Question: When do I multiply by the decimal and when do I divide by it? I keep mixing them up.

Answer: The operation depends on what you're trying to find:

Finding a Part (most common):

  • Formula: Part = Whole × Decimal
  • Example: What is 25% of 80? → 80 × 0.25 = 20
  • Multiply the whole by the decimal

Finding the Whole:

  • Formula: Whole = Part ÷ Decimal
  • Example: 20 is 25% of what? → 20 ÷ 0.25 = 80
  • Divide the part by the decimal

Finding the Percent:

  • Formula: Percent = (Part ÷ Whole) × 100
  • Example: What percent of 80 is 20? → (20 ÷ 80) × 100 = 25%

Remember: Part = Whole × Percent (in decimal form)

Question: I sometimes get decimal answers when solving percent problems. Is this normal? What if I get 0.45 or 1.23?

Answer: Yes, decimal answers are completely normal in percent calculations! Many real-world problems result in decimal answers.

Types of decimal results:

  • 0.45 could mean 45% when converted back to percent
  • 1.23 could mean a 123% result (more than the whole)
  • 0.08 could mean 8% of something

When decimals make sense:

  • Percentages like 45.7%, 12.3%
  • Money amounts: $24.75
  • Measurements: 3.6 kg, 1.8 meters

When dealing with decimals:

  • Round appropriately based on context (money to nearest cent)
  • Keep more precision during calculations, round final answer
  • Check that your answer makes sense in the real-world context

Decimals often occur because real-world measurements rarely divide evenly!